Error estimates for semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth data
type: Item
instance of: scholarly article
title: Error estimates for semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth data
zbMATH Open document ID: 1012.65149
published in: Calcolo
publication date: 2003-03-18T00:00:00Z
review text: The paper deals with a parabolic integro-differential equation of the form \[ u_t + A(t)u = \int_0^t B(t,s) u(s) ds + f \tag{1} \] in \(\Omega \times (0,T]\) subjected to the boundary and initial conditions \[ u = 0 \quad \text{on} \quad \partial \Omega \times (0,T], \qquad u(\cdot,0) = u_0 \quad \text{in} \quad \Omega. \tag{2} \] Here \(\Omega \subset R^d\) is a bounded domain with smooth boundary, \(T < \infty\). The coefficients of the second-order linear partial differential operators \(A(t), B(t,s)\) are smooth, moreover \(A(t)\) is supposed to be a selfadjoint and uniformly positive definite elliptic operator. Integro-differential equations of the above form occur in many applications as heat conduction in materials with memory, compression of poroviscoelastic media, and nuclear reactor dynamics. The paper was stimulated by an open problem posed by \textit{C. Chen} and \textit{T. Shih} [Finite element methods for integro-differential equations, Singapore (1998; Zbl 0909.65142)]. Introducing a weak formulation of (1), (2) the authors consider its finite element Galerkin approximation with respect to the space variable (with continuous time \(t \in (0,T]\)). First, a priori estimates for the exact solution are obtained, using standard energy methods and a suitably defined Ritz-Volterra projection. In the sequel the error estimates are proved in the following two cases: (i) \(f=0\), \(u_0 \in H^2(\Omega)\) or \(u_0 \in L^2(\Omega)\) and (ii) \(f(t) \in L^2\Omega)\), \(u_0 = 0\).
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 80A20
Mathematics Subject Classification ID: 80M10
Mathematics Subject Classification ID: 82D75
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 74D05
Mathematics Subject Classification ID: 74S05
zbMATH DE Number: 1884270
zbMATH Keywords: parabolic integro-differential equations
zbMATH Keywords: Galerkin approximation
zbMATH Keywords: finite element approximation
zbMATH Keywords: Ritz-Volterra projection
zbMATH Keywords: error estimates
zbMATH Keywords: heat conduction in materials with memory
zbMATH Keywords: compression of poroviscoelastic media
zbMATH Keywords: nuclear reactor dynamics
author: Amiya K. Pani
author: Rajen Kumar Sinha
reviewed by: Hanna Marcinkowska
MaRDI profile type: Publication
full work available at URL: s100920070001
OpenAlex ID: W2009477751
Wikidata QID: Q115385389
DOI: 10.1007/S100920070001
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description: scientific article; zbMATH DE number 1884270